Question

Tunjukkan bahwa i^2 = n (n + 1) (2n + 1)/ 6 jika n adalah suatu bilangan bulat positif.

Answer 1

Misalkan,
i1 = 1
i2 = 1+4 = 5
i3 = 1+4+9 = 14
i4 = 1+4+9+16 = 30
dst.
Hingga
1....5....14....30
...4....9.....16
......5....7
.........2
Barisan bilangan berpangkat tiga,
Un = an^3+bn^2+cn+d
Sehingga
U1 = a+b+c+d
U2 = 8a+4b+2c+d
dst.
a+b+c+d......8a+4b+2c+d......27a+9b+3c+d......64a+16b+4c+d
..........7a+3b+c..........19a+5b+c..........37a+7b+c
.....................12a+2b...............18a+2b
.....................................6a
Dari situ didapat:
6a = 2
a = 1/3,

12a+2b = 5
4+2b = 5
b = 1/2

7a+3b+c = 4
7/3 + 3/2 + c = 4
c = 1/6

a+b+c+d = 1
1/3+1/2+1/6+d = 1
d = 0

Didapat Un = 1/3 n^3 + 1/2 n^2 +1/6 n
1/6 (2n^3 + 3n^2 + n)
1/6 n(2n^2+3n+1)
1/6 n(n+1)(2n+1)
n(n+1)(2n+1)/6